Jeremy Corbyn was drawn into an extraordinary row with Virgin Trains after the rail operator disputed the Labour leader’s filmed account of having to sit on the floor of a “ram-packed” train, releasing CCTV images of him walking past free seats beforehand and sitting down shortly after the film was shot.

Backed up by Virgin Group founder Sir Richard Branson, the train operator said it had to “take issue with the idea that Mr Corbyn wasn’t able to be seated on the service, as this clearly wasn’t the case” on a crowded train that the Labour leader had taken from London to Newcastle on 11 August.

But the Labour leader stood by his description of the train journey, saying while there had been some available seats he had not been able to sit with his wife, Laura Alvarez, and that he was only able to sit later because train staff had upgraded another family to first class to create space.

Virgin Trains took the unusual step just more than a week after Corbyn was filmed sitting on the train floor after he could not find a seat on the morning train. In the video he laments the lack of available seats, saying: “This is a problem that many passengers face every day; commuters and long-distance travellers. Today this train is completely ram-packed.”

One of the images released by Virgin Trains, which shows a time code of 11.10am – 10 minutes after the service had departed from London – showed Corbyn walking past seemingly unoccupied seats which did not have slips attached to them indicating they were reserved. Virgin Trains also released another CCTV still, from 11.43am, about 15 minutes after the video was shot. This showed Corbyn and his travelling companions returning to the same carriage to sit down.

A Corbyn source said a handful of seats had been available, but not two together, and the Labour leader wanted to sit with his wife. As a result he, his wife and team were among a series of other passengers forced to sit or stand in the corridor.

The source added that when Virgin staff spotted Corbyn sitting in a vestibule they offered him a complimentary upgrade to first class, which he refused.

He added: “Seats became available after a family were upgraded to first class, and Jeremy and the team he was travelling with were offered the seats by a very helpful member of staff.”

Corbyn’s spokesman sought to tie the row to the Labour leader’s wider transport policies. “Passengers across Britain will have been in similar situations on overcrowded, expensive trains,” he said.

“That is why our policy to bring the trains back into public ownership, as part of a plan to rebuild and transform Britain, is so popular with passengers and rail workers.”

Branson also tweeted to his 8.2 million followers a third image of Corbyn walking through a crowded carriage, timed at 11.11am, where the seats were clearly marked with reservation tickets. The Virgin entrepreneur wrote: “Mr Corbyn & team walked past empty-unreserved seats then filmed claim train was ‘ram-packed’”.

Yannis Mendez, who filmed the original video – he has been following Corbyn and volunteers for his leadership campaign against Owen Smith – added that some of the seemingly empty seats shown in the first Virgin Trains image had bags and coats on them, so were not free.

The Corbyn team’s account was supported by a woman who said she sat on the floor next to the Labour leader, having sent a social media photo of herself and her daughter with him.

Ellen – who asked to not be named in full– told the Guardian that Corbyn had seemingly gone through the entire standard-class section of the train but had not been able to find a seat. The 26-year-old, who was with her one-year-old daughter and son, aged six, said she had similarly been unable to find seats.

“He’s not lying,” she said. “When I saw him he was in coach A, right at the front. He hadn’t managed to find a seat in the whole of the train. I was sat on the floor, there was no space for me to get a seat. There were people in every space between every carriage. It was totally overcrowded. They were full of bags and full of people.”

Another woman, Keren Harrison, tweeted a photo of herself with Corbyn on the train, saying there was only a seat for him about 45 minutes into the three-hour trip “when staff started shuffling people around”. This process appeared to involve Virgin staff directing other passengers sitting in corridors to reserved seats which had not been occupied.

A spokesperson for Virgin Trains said: “We know that some of our services on our east and west coast franchises are extremely popular, and it can be hard to find a seat.” It usually happened in particular circumstances, the operator added, for example when there’s a big sporting event, or on the first off-peak train out of London.

“Unfortunately, we can’t do anything about cup finals or fares regulation, which could spread demand much more effectively if it was less of a blunt instrument,” the spokesperson said, arguing they would be “delighted to work with ministers if they were interested in reviewing the fares structure for long distance services, with the aim of reducing the overcrowding that can occur”.

At a rally in east London on Tuesday night night, Sam Tarry, the Labour leader’s campaign director, told the crowds that Branson had released the letter because he was upset about Corbyn’s plans to renationalise the railways. “Some of you might have seen on social media today there’s been a little bit of a spat,” he said. “Richard Branson has decided he’s very upset about our not particularly radical plans to renationalise our railways, so he’s having a little pop at us.

“I’d just say that’s very, very indicative – the establishment is absolutely petrified about what this campaign is about, what this movement is about.”

Smith, who is challenging Corbyn for the Labour leadership, was quick to poke fun at the situation, tweeting: “My campaign remains on track. Proud to be genuinely standing up for ordinary people.”

Owen Smith (@OwenSmith_MP)

My campaign remains on track. Proud to be genuinely standing up for ordinary people.

Smith will try to regain the initiative in the leadership contest today with a pledge to block Brexit in parliament.

The Pontypridd MP will say a Labour party led by him would vote against the triggering of article 50 – the formal process for Brexit – until the Conservatives agreed to put the final negotiated departure to the people, either through a referendum or at a general election.

The track record: how the journey unfolded

11.10amJeremy Corbyn and his wife, Laura Alvarez, board the train. Virgin Trains CCTV shows seemingly unoccupied seats without reservation slips. A source close to the Labour leader said there were not enough free seats for Corbyn to sit with his wife.

11.11am A second CCTV image, with a time stamp of 11.11am, tweeted by Virgin’s owner, Richard Branson, shows Corbyn clearly walking past empty seats. However, these all seem to have slips indicating that they are reserved.

11.30am A few minutes later, Yannis Mendez, a freelance filmmaker accompanying Corbyn, films him sitting on the floor. He says the ‘ram-packed’ train’s lack of seats ­emphasises the need for the railways to be returned to public ownership.

11.43am Corbyn takes a seat in coach H after staff directed passengers to use booked seats that had not been taken. A Corbyn source said the leader’s seats were provided after Virgin staff upgraded a family to first class to make space for him and Alvarez.